Teacher Overview

CCGPS
Frameworks
Teacher Edition

Mathematics

First Grade
Grade Level Overview

Georgia Department of Education
Common Core Georgia Performance Standards Framework
First Grade Mathematics • Grade Level Overview

Grade Level Overview

TABLE OF CONTENTS

Curriculum Map and pacing Guide..................................................................................................3

Unpacking the Standards ...................................................................................................................

• Standards of Mathematical Practice.....................................................................................4

• Content Standards ................................................................................................................6

Arc of Lesson/Math Instructional Framework ...............................................................................23

Unpacking a Task ..........................................................................................................................24

Routines and Rituals ......................................................................................................................25

General Questions for Teacher Use ...............................................................................................34

Questions for Teacher Reflection ..................................................................................................35

Depth of Knowledge ......................................................................................................................36

Depth and Rigor Statement ............................................................................................................38

Additional Resources Available ....................................................................................................39

• K-2 Problem Solving Rubric .............................................................................................39

• Literature Resources ..........................................................................................................41

• Technology Links ..............................................................................................................41

Resources Consulted ......................................................................................................................43

MATHEMATICS GRADE 1 Grade Level Overview
Dr. John D. Barge, State School Superintendent
April 2012 Page 2 of 43
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Common Core Georgia Performance Standards
First Grade
Common Core Georgia Performance Standards: Curriculum Map
Unit 1 Unit 2 Unit 3 Unit 4 Unit 5 Unit 6 Unit 7
Creating Developing Base Understanding Sorting, Operations and Understanding Show What We Know
Routines Using Ten Number Shapes and Comparing Algebraic Thinking Place Value
Data Sense Fractions and Ordering
MCC1.NBT.1 MCC1.NBT.1 MCC1.G.1 MCC1.MD.1 MCC1.OA.1 MCC1.NBT.2 ALL
MCC1.MD.4 MCC1.MD.4 MCC1.G.2 MCC1.MD.2 MCC1.OA.2 MCC1.NBT.3
MCC1.G.3 MCC1.MD.3 MCC1.OA.3 MCC1.NBT.4
MCC1.MD.4 MCC1.MD.4 MCC1.OA.4 MCC1.NBT.5
MCC1.OA.5 MCC1.NBT.6
MCC1.OA.6 MCC1.MD.4
MCC1.OA.7
MCC1.OA.8
MCC1.MD.4

These units were written to build upon concepts from prior units, so later units contain tasks that depend upon the concepts addressed in earlier units.
All units will include the Mathematical Practices and indicate skills to maintain.
NOTE: Mathematical standards are interwoven and should be addressed throughout the year in as many different units and tasks as possible in order to stress the natural connections that exist among
mathematical topics.

Grades K-2 Key: CC = Counting and Cardinality, G= Geometry, MD=Measurement and Data, NBT= Number and Operations in Base Ten, OA = Operations and Algebraic Thinking.

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STANDARDS OF MATHEMATICAL PRACTICE

The Standards for Mathematical Practice describe varieties of expertise that mathematics
educators at all levels should seek to develop in their students. These practices rest on important
“processes and proficiencies” with longstanding importance in mathematics education.

The first of these are the NCTM process standards of problem solving, reasoning and
proof, communication, representation, and connections.

The second are the strands of mathematical proficiency specified in the National
Research Council’s report Adding It Up: adaptive reasoning, strategic competence,
conceptual understanding (comprehension of mathematical concepts, operations and
relations), procedural fluency (skill in carrying out procedures flexibly, accurately,
efficiently and appropriately), and productive disposition (habitual inclination to see
mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and
one’s own efficacy).

Students are expected to:
1. Make sense of problems and persevere in solving them.
In first grade, students realize that doing mathematics involves solving problems and discussing
how they solved them. Students explain to themselves the meaning of a problem and look for
ways to solve it. Younger students may use concrete objects or pictures to help them
conceptualize and solve problems. They may check their thinking by asking themselves, “Does
this make sense?” They are willing to try other approaches.

2. Reason abstractly and quantitatively.
Younger students recognize that a number represents a specific quantity. They connect the
quantity to written symbols. Quantitative reasoning entails creating a representation of a problem
while attending to the meanings of the quantities.

3. Construct viable arguments and critique the reasoning of others.
First graders construct arguments using concrete referents, such as objects, pictures, drawings,
and actions. They also practice their mathematical communication skills as they participate in
mathematical discussions involving questions like “How did you get that?” “Explain your
thinking,” and “Why is that true?” They not only explain their own thinking, but listen to others’
explanations. They decide if the explanations make sense and ask questions.

4. Model with mathematics.
In early grades, students experiment with representing problem situations in multiple ways
including numbers, words (mathematical language), drawing pictures, using objects, acting out,
making a chart or list, creating equations, etc. Students need opportunities to connect the
different representations and explain the connections. They should be able to use all of these
representations as needed.

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5. Use appropriate tools strategically.
In first grade, students begin to consider the available tools (including estimation) when solving
a mathematical problem and decide when certain tools might be helpful. For instance, first
graders decide it might be best to use colored chips to model an addition problem.

6. Attend to precision.
As young children begin to develop their mathematical communication skills, they try to use
clear and precise language in their discussions with others and when they explain their own
reasoning.

7. Look for and make use of structure.
First graders begin to discern a pattern or structure. For instance, if students recognize 12 + 3 =
15, then they also know 3 + 12 = 15. (Commutative property of addition.) To add 4 + 6 + 4, the
first two numbers can be added to make a ten, so 4 + 6 + 4 = 10 + 4 = 14.

8. Look for and express regularity in repeated reasoning.
In the early grades, students notice repetitive actions in counting and computation, etc. When
children have multiple opportunities to add and subtract “ten” and multiples of “ten” they notice
the pattern and gain a better understanding of place value. Students continually check their work
by asking themselves, “Does this make sense?”

***Mathematical Practices 1 and 6 should be evident in EVERY lesson***

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CONTENT STANDARDS

OPERATIONS AND ALGEBRAIC THINKING (OA)

CLUSTER #1: REPRESENT AND SOLVE PROBLEMS INVOLVING ADDITION AND
SUBTRACTION.
Students develop strategies for adding and subtracting whole numbers based on their prior work
with small numbers. They use a variety of models, including discrete objects and length-based
models (e.g., cubes connected to form lengths), to model add-to, take-from, put-together, take-
apart, and compare situations to develop meaning for the operations of addition and subtraction,
and to develop strategies to solve arithmetic problems with these operations. Prior to first grade
students should recognize that any given group of objects (up to 10) can be separated into sub
groups in multiple ways and remain equivalent in amount to the original group (Ex: A set of 6
cubes can be separated into a set of 2 cubes and a set of 4 cubes and remain 6 total cubes).

CCGPS.1.OA.1 Use addition and subtraction within 20 to solve word problems involving
situations of adding to, taking from, putting together, taking apart, and comparing, with
unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for
the unknown number to represent the problem.
This standard builds on the work in Kindergarten by having students use a variety of
mathematical representations (e.g., objects, drawings, and equations) during their work.
The unknown symbols should include boxes or pictures, and not letters.
Teachers should be cognizant of the three types of problems. There are three types of
addition and subtraction problems: Result Unknown, Change Unknown, and Start
Unknown. Here are some Addition
Use informal language (and, minus/subtract, the same as) to describe joining situations
(putting together) and separating situations (breaking apart).
Use the addition symbol (+) to represent joining situations, the subtraction symbol (-) to
represent separating situations, and the equal sign (=) to represent a relationship
regarding quantity between one side of the equation and the other.
A helpful strategy is for students to recognize sets of objects in common patterned
arrangements (0-6) to tell how many without counting (subitizing).
Examples:
Result Unknown Change Unknown Start Unknown
There are 9 students on the There are 9 students on the There are some students on the
playground. Then 8 more playground. Some more playground. Then 8 more
students showed up. How students show up. There are students came. There are now
many students are there now? now 17 students. How many 17 students. How many
(9 + 8 = ____) students came? (9 + ____ = students were on the
17) playground at the beginning?
(____ + 8 = 17)

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CCGPS.1.OA.2 Solve word problems that call for addition of three whole numbers whose
sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a
symbol for the unknown number to represent the problem.
This standard asks students to add (join) three numbers whose sum is less than or equal to
20, using a variety of mathematical representations.
This objective does address multi-step word problems.
Example:
There are cookies on the plate. There are 4 oatmeal raisin cookies, 5 chocolate chip
cookies, and 6 gingerbread cookies. How many cookies are there total?
Student 1: Adding with a Ten Frame and Counters
I put 4 counters on the Ten Frame for the oatmeal raisin cookies. Then I put 5 different
color counters on the ten-frame for the chocolate chip cookies. Then I put another 6 color
counters out for the gingerbread cookies. Only one of the gingerbread cookies fit, so I had 5
leftover. One ten and five leftover makes 15 cookies.

Student 2: Look for Ways to Make 10
I know that 4 and 6 equal 10, so the oatmeal raisin and gingerbread equals 10 cookies. Then
I add the 5 chocolate chip cookies and get 15 total cookies.
Student 3: Number Line
I counted on the number line. First I counted 4, and then I counted 5 more and landed on 9.
Then I counted 6 more and landed on 15. So there were 15 total cookies.

CLUSTER #2: UNDERSTAND AND APPLY PROPERTIES OF OPERATIONS AND
THE RELATIONSHIP BETWEEN ADDITION AND SUBTRACTION.
Students understand connections between counting and addition and subtraction (e.g., adding
two is the same as counting on two). They use properties of addition to add whole numbers and
to create and use increasingly sophisticated strategies based on these properties (e.g., “making
tens”) to solve addition and subtraction problems within 20. By comparing a variety of solution
strategies, children build their understanding of the relationship between addition and
subtraction.

CCGPS.1.OA.3 Apply properties of operations as strategies to add and subtract.
Examples: If 8 + 3 = 11 is known, then 3 + 8 = 11 is also known. (Commutative property of
addition.) To add 2 + 6 + 4, the second two numbers can be added to make a ten, so 2 + 6 +
4 = 2 + 10 = 12. (Associative property of addition.)
This standard calls for students to apply properties of operations as strategies to add and
subtract. Students do not need to use formal terms for these properties. Students should

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use mathematical tools, such as cubes and counters, and representations such as the
number line and a 100 chart to model these ideas.
Example:
Student can build a tower of 8 green cubes and 3 yellow cubes and another tower of 3
yellow and 8 green cubes to show that order does not change the result in the operation of
addition. Students can also use cubes of 3 different colors to “prove” that (2 + 6) + 4 is
equivalent to 2 + (6 + 4) and then to prove 2 + 6 + 4 = 2 + 10.
Commutative Property of Addition Associative Property of Addition
Order does not matter when you add When adding a string of numbers you can add
numbers. For example, if 8 + 2 = 10 is any two numbers first. For example, when
known, then 2 + 8 = 10 is also known. adding 2 + 6 + 4, the second two numbers can
be added to make a ten, so 2+6+
4 = 2 + 10 = 12

Student Example: Using a Number Balance to Investigate the Commutative Property
If I put a weight on 8 first and then 2, I think that will balance if I put a weight on 2 first this
time and then on 8.

CCGPS.1.OA.4 Understand subtraction as an unknown-addend problem. For example,
subtract 10 – 8 by finding the number that makes 10 when added to 8. Add and subtract
within 20.
This standard asks for students to use subtraction in the context of unknown addend
problems. Example: 12 – 5 = __ could be expressed as 5 + __ = 12. Students should use
cubes and counters, and representations such as the number line and the100 chart, to
model and solve problems involving the inverse relationship between addition and
subtraction.

Student 1 Student 2
I used a ten-frame. I started with 5 counters. I I used a part-part-whole diagram. I put 5 counters
knew that I had to have 12, which is one full ten on one side. I wrote 12 above the diagram. I put
frame and two leftovers. I needed 7 counters, so counters into the other side until there were 12 in
12 – 5 = 7. all. I know I put 7 counters on the other side, so 12
– 5 = 7.

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Student 3: Draw a Number Line
I started at 5 and counted up until I reached 12. I counted 7 numbers, so I know that 12 – 5 = 7.

CLUSTER #3: ADD AND SUBTRACT WITHIN 20.

CCGPS.1.OA.5 Relate counting to addition and subtraction (e.g., by counting on 2 to add
2).
This standard asks for students to make a connection between counting and adding and
subtraction. Students use various counting strategies, including counting all, counting on,
and counting back with numbers up to 20. This standard calls for students to move
beyond counting all and become comfortable at counting on and counting back. The
counting all strategy requires students to count an entire set. The counting and counting
back strategies occur when students are able to hold the ―start number‖ in their head and
count on from that number.
Example: 5 + 2 = ___
Student 1: Counting All Student 2: Counting On
5 + 2 = ___. The student counts five 5 + 2 = ___. Student counts five counters.
counters. The student adds two more. The student adds the first counter and says 6,
The student counts 1, 2, 3, 4, 5, 6, 7 to then adds another counter and says 7. The
get the answer. student knows the answer is 7, since they
counted on 2.
Example: 12 – 3 = ___
Student 1: Counting All Student 2: Counting Back
12 – 3 = ___. The student counts twelve 12 – 3 = ___. The student counts twelve
counters. The student removes 3 of counters. The student removes a counter and
them. The student counts 1, 2, 3, 4, 5, 6, says 11, removes another counter and says 10,
7, 8, 9 to get the answer. and removes a third counter and says 9. The
student knows the answer is 9, since they
counted back 3.

CCGPS.1.OA.6 Add and subtract within 20, demonstrating fluency for addition and
subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 +
4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 =
9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12,
one knows 12 – 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7
by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).
This standard mentions the word fluency when students are adding and subtracting
numbers within 10. Fluency means accuracy (correct answer), efficiency (within 4-5
seconds), and flexibility (using strategies such as making 5 or making 10).
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The standard also calls for students to use a variety of strategies when adding and
subtracting numbers within 20. Students should have ample experiences modeling these
operations before working on fluency. Teacher could differentiate using smaller
numbers.
It is importance to move beyond the strategy of counting on, which is considered a less
important skill than the ones here in 1.OA.6. Many times teachers think that counting on
is all a child needs, when it is really not much better skill than counting all and can
becomes a hindrance when working with larger numbers.

Example: 8 + 7 = ___
Student 1: Making 10 and Decomposing a Student 2: Creating an Easier Problem
Number with Known Sums
I know that 8 plus 2 is 10, so I decomposed I know 8 is 7 + 1. I also know that 7 and 7
(broke) the 7 up into a 2 and a 5. First I equal 14 and then I added 1 more to get 15.
added 8 and 2 to get 10, and then added the 8 + 7 = (7 + 7) + 1 = 15
5 to get 15.
8 + 7 = (8 + 2) + 5 = 10 + 5 = 15

Example: 14 – 6 = ___
Student 1: Decomposing the Number You Student 2: Relationship between Addition
Subtract and Subtraction
I know that 14 minus 4 is 10 so I broke the 6 + is 14. I know that 6 plus 8 is 14, so
6 up into a 4 and a 2. 14 minus 4 is 10. that means that 14 minus 6 is 8.
Then I take away 2 more to get 8. 6 + 8 = 14 so 14 – 6 = 8
14 – 6 = (14 – 4) – 2 = 10 – 2 = 8

Algebraic ideas underlie what students are doing when they create equivalent expressions
in order to solve a problem or when they use addition combinations they know to solve
more difficult problems. Students begin to consider the relationship between the parts.
For example, students notice that the whole remains the same, as one part increases the
other part decreases. 5 + 2 = 4 + 3

CLUSTER #4: WORK WITH ADDITION AND SUBTRACTION EQUATIONS.

CCGPS.1.OA.7 Understand the meaning of the equal sign, and determine if equations
involving addition and subtraction are true or false. For example, which of the following
equations are true and which are false? 6 = 6, 7 = 8 – 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.
This standard calls for students to work with the concept of equality by identifying
whether equations are true or false. Therefore, students need to understand that the equal
sign does not mean ―answer comes next‖, but rather that the equal sign signifies a
relationship between the left and right side of the equation.
The number sentence 4 + 5 = 9 can be read as, ―Four plus five is the same amount as
nine.‖ In addition, Students should be exposed to various representations of equations,
such as: an operation on the left side of the equal sign and the answer on the right side (5
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+ 8 = 13) an operation on the right side of the equal sign and the answer on the left side
(13 = 5 + 8) numbers on both sides of the equal sign (6 = 6) operations on both sides of
the equal sign (5 + 2 = 4 + 3). Students need many opportunities to model equations
using cubes, counters, drawings, etc.

CCGPS.1.OA.8 Determine the unknown whole number in an addition or subtraction
equation relating three whole numbers. For example, determine the unknown number that
makes the equation true in each of the equations 8 + ? = 11, 5 = _ – 3, 6 + 6 = _.
This standard extends the work that students do in 1.OA.4 by relating addition and
subtraction as related operations for situations with an unknown. This standard builds
upon the ―think addition‖ for subtraction problems as explained by Student 2 in
CCGPS.1.OA.6.
Student 1
5 = ___ – 3
I know that 5 plus 3 is 8. So 8 minus 3 is
5.

NUMBERS AND OPERATIONS IN BASE TEN (NBT)

CLUSTER #1: EXTEND THE COUNTING SEQUENCE.

CCGPS.1.NBT.1 Count to 120, starting at any number less than 120. In this range, read
and write numerals and represent a number of objects with a written numeral.
This standard calls for students to rote count forward to 120 by Counting On from any
number less than 120. Students should have ample experiences with the hundreds chart
to see patterns between numbers, such as all of the numbers in a column on the hundreds
chart have the same digit in the ones place, and all of the numbers in a row have the same
digit in the tens place.
This standard also calls for students to read, write and represent a number of objects with
a written numeral (number form or standard form). These representations can include
cubes, place value (base 10) blocks, pictorial representations or other concrete materials.
As students are developing accurate counting strategies they are also building an
understanding of how the numbers in the counting sequence are related—each number is
one more (or one less) than the number before (or after).

CLUSTER#2: UNDERSTAND PLACE VALUE.
Students develop, discuss, and use efficient, accurate, and generalizable methods to add within
100 and subtract multiples of 10. They compare whole numbers (at least to 100) to develop
understanding of and solve problems involving their relative sizes. They think of whole numbers
between 10 and 100 in terms of tens and ones (especially recognizing the numbers 11 to 19 as
composed of a ten and some ones). Through activities that build number sense, they understand
the order of the counting numbers and their relative magnitudes.

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CCGPS.1.NBT.2 Understand that the two digits of a two-digit number represent amounts of
tens and ones. Understand the following as special cases:

a. 10 can be thought of as a bundle of ten ones – called a “ten.”
This standard asks students to unitize a group of ten ones as a whole unit: a ten. This is
the foundation of the place value system. So, rather than seeing a group of ten cubes as
ten individual cubes, the student is now asked to see those ten cubes as a bundle – one
bundle of ten.

b. The numbers from 11 to 19 are composed of a ten and one, two, three, four, five,
six, seven, eight, or nine ones.
This standard asks students to extend their work from Kindergarten when they composed
and decomposed numbers from 11 to 19 into ten ones and some further ones. In
Kindergarten, everything was thought of as individual units: ―ones‖. In First Grade,
students are asked to unitize those ten individual ones as a whole unit: ―one ten‖.
Students in first grade explore the idea that the teen numbers (11 to 19) can be expressed
as one ten and some leftover ones. Ample experiences with ten frames will help develop
this concept.
Example:
For the number 12, do you have enough to make a ten? Would you have any leftover? If
so, how many leftovers would you have?

Student 1: Student 2:
I filled a ten-frame to make one ten and I counted out 12 place value cubes. I
had two counters left over. I had enough had enough to trade 10 cubes for a ten-
to make a ten with some left over. The rod (stick). I now have 1 ten-rod and 2
number 12 has 1 ten and 2 ones. cubes left over. So the number 12 has
1 ten and 2 ones.

c. The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five,
six, seven, eight, or nine tens (and 0 ones).
This standard builds on the work of CCGPS.1.NBT.2b. Students should explore the idea
that decade numbers (e.g., 10, 20, 30, 40) are groups of tens with no left over ones.
Students can represent this with cubes or place value (base 10) rods. (Most first grade
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students view the ten stick (numeration rod) as ONE. It is recommended to make a ten
with unfix cubes or other materials that students can group. Provide students with
opportunities to count books, cubes, pennies, etc. Counting 30 or more objects supports
grouping to keep track of the number of objects.)

CCGPS.1.NBT.3 Compare two two-digit numbers based on meanings of the tens and ones
digits, recording the results of comparisons with the symbols >, =, and <.
This standard builds on the work of CCGPS.1.NBT.1 and CCGPS.1.NBT.2 by having
students compare two numbers by examining the amount of tens and ones in each
number. Students are introduced to the symbols greater than (>), less than (<) and equal
to (=). Students should have ample experiences communicating their comparisons using
words, models and in context before using only symbols in this standard.
Example: 42 ___ 45

42 has 4 tens and 2 ones. 45 has 4 tens 42 is less than 45. I know this because
and 5 ones. They have the same number when I count up I say 42 before I say 45.
of tens, but 45 has more ones than 42. So So, 42 < 45.
45 is greater than 42. So, 42 < 45.

CLUSTER #4: USE PLACE VALUE UNDERSTANDING AND PROPERTIES OF
OPERATIONS TO ADD AND SUBTRACT.

CCGPS.1.NBT.4 Add within 100, including adding a two-digit number and a one-digit
number, and adding a two-digit number and a multiple of 10, using concrete models or
drawings and strategies based on place value, properties of operations, and/or the
relationship between addition and subtraction; relate the strategy to a written method and
explain the reasoning used. Understand that in adding two-digit numbers, one adds tens
and tens, ones and ones; and sometimes it is necessary to compose a ten.
This standard calls for students to use concrete models, drawings and place value
strategies to add and subtract within 100. Students should not be exposed to the standard
algorithm of carrying or borrowing in first grade.
Example:

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There are 37 children on the playground. When a class of 23 students come to the
playground, how many students are on the playground altogether?

Student 1
I used a hundreds chart. I started at 37 and moved
over 3 to land on 40. Then to add 20 I moved down 2
rows and landed on 60. So there are 60 people on the
playground.

Student 2
I used place value blocks and made a pile of 37 and a
pile of 23. I joined the tens and got 50. I then joined
the ones and got 10. I then combined those piles and
got 60. So there are 60 people on the playground.
Relate models to symbolic notation.

Student 3
I broke 37 and 23 into tens and ones. I added the tens and got
50. I added the ones and got 10. I know that 50 and 10 more
is 60. So, there are 60 people on the playground. Relate
models to symbolic notation.

Student 4
Using mental math, I started at 37 and counted on 3 to get 40. Then I added 20 which is 2
tens, to land on 60. So, there are 60 people on the playground.

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Student 5
I used the number line. I started at 37. Then I broke up 23 into 20 and 3 in my head. Next, I
added 3 ones to get to 40. I then jumped 10 to get to 50 and 10 more to get to 60. So there
are 60 people on the playground.

CCGPS.1.NBT.5 Given a two-digit number, mentally find 10 more or 10 less than the
number, without having to count; explain the reasoning used.
This standard builds on students’ work with tens and ones by mentally adding ten more
and ten less than any number less than 100. Ample experiences with ten frames and the
hundreds chart help students use the patterns found in the tens place to solve such
problems.
Example:
There are 74 birds in the park. 10 birds fly away. How many are left?

Student 1
I used a 100s board. I started at 74. Then, because
10 birds flew away, I moved back one row. I landed
on 64. So, there are 64 birds left in the park.

Student 2
I pictured 7 ten-frames and 4 left over in my head. Since 10
birds flew away, I took one of the ten-frames away. That left
6 ten-frames and 4 left over. So, there are 64 birds left in the
park.

CCGPS.1.NBT.6 Subtract multiples of 10 in the range 10-90 from multiples of 10 in the
range 10-90 (positive or zero differences), using concrete models or drawings and strategies
based on place value, properties of operations, and/or the relationship between addition
and subtraction; relate the strategy to a written method and explain the reasoning used.
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This standard calls for students to use concrete models, drawings and place value
strategies to subtract multiples of 10 from decade numbers (e.g., 30, 40, 50).
Example:
There are 60 students in the gym. 30 students leave. How many students are still in the
gym?

Student 1
I used a 100s chart and started at 60. I moved up 3 rows to
land on 30. There are 30 students left.

Student 2
I used place value blocks or unifix cubes to build towers of 10. I
started with 6 towers of 10 and removed 3 towers. I had 3 towers
left. 3 towers have a value of 30. So there are 30 students left.

Student 3
Using mental math, I solved this subtraction problem. I know that 30 plus 30 is 60, so 60
minus 30 equals 30. There are 30 students left..

Student 4
I used a number line. I started with 60 and moved back 3 jumps of 10 and landed on 30.
There are 30 students left.

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MEASUREMENT AND DATA (MD)

CCGPS CLUSTER #1: MEASURE LENGTHS INDIRECTLY AND BY ITERATING
LENGTH UNITS.
Students develop an understanding of the meaning and processes of measurement, including
underlying concepts such as iterating (the mental activity of building up the length of an object
with equal-sized units) and the transitivity principle for indirect measurement.1
1Students should apply the principle of transitivity of measurement to make indirect
comparisons, but they need not use this technical term.

CCGPS.1.MD.1 Order three objects by length; compare the lengths of two objects
indirectly by using a third object.
This standard calls for students to indirectly measure objects by comparing the length of
two objects by using a third object as a measuring tool. This concept is referred to as
transitivity.
Example:
Which is longer: the height of the bookshelf or the height of a desk?
I used a pencil to measure the height of the I used a book to measure the bookshelf and
bookshelf and it was 6 pencils long. I used it was 3 books long. I used the same book
the same pencil to measure the height of the to measure the height of the desk and it was
desk and the desk was 4 pencils long. a little less than 2 books long. Therefore,
Therefore, the bookshelf is taller than the the bookshelf is taller than the desk.
desk.

CCGPS.1.MD.2 Express the length of an object as a whole number of length units, by
laying multiple copies of a shorter object (the length unit) end to end; understand that the
length measurement of an object is the number of same-size length units that span it with
no gaps or overlaps. Limit to contexts where the object being measured is spanned by a whole
number of length units with no gaps or overlaps.
This standard asks students to use multiple copies of one object to measure a larger
object. This concept is referred to as iteration. Through numerous experiences and
careful questioning by the teacher, students will recognize the importance of making sure
that there are not any gaps or overlaps in order to get an accurate measurement. This
concept is a foundational building block for the concept of area in 3rd Grade.
Example:
How long is the paper in terms of paper clips?

CCGPS CLUSTER #2: TELL AND WRITE TIME.

CCGPS.1.MD.3 Tell and write time in hours and half-hours using analog
and digital clocks.
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This standard calls for students to read both analog and digital clocks and then orally tell
and write the time. Times should be limited to the hour and the half-hour. Students need
experiences exploring the idea that when the time is at the half-hour the hour hand is
between numbers and not on a number. Further, the hour is the number before where the
hour hand is. For example, in the clock at the right, the time is 8:30. The hour hand is
between the 8 and 9, but the hour is 8 since it is not yet on the 9.

CCGPS CLUSTER #3: REPRESENT AND INTERPRET DATA.

CCGPS.1.MD.4 Organize, represent, and interpret data with up to three categories; ask
and answer questions about the total number of data points, how many in each category,
and how many more or less are in one category than in another.
This standard calls for students to work with categorical data by organizing, representing
and interpreting data. Students should have experiences posing a question with 3
possible responses and then work with the data that they collect. For example:
Students pose a question and the 3 possible responses: Which is your favorite flavor of
ice cream? Chocolate, vanilla or strawberry? Students collect their data by using tallies
or another way of keeping track. Students organize their data by totaling each category in
a chart or table. Picture and bar graphs are introduced in 2nd Grade.

What is your favorite flavor of ice cream?
Chocolate 12
Vanilla 5
Strawberry 6

Students interpret the data by comparing categories.
Examples of comparisons:
• What does the data tell us? Does it answer our question?
• More people like chocolate than the other two flavors.
• Only 5 people liked vanilla.
• Six people liked Strawberry.
• 7 more people liked Chocolate than Vanilla.
• The number of people that liked Vanilla was 1 less than the number of people
who liked Strawberry.
• The number of people who liked either Vanilla or Strawberry was 1 less than the
number of people who liked chocolate.
• 23 people answered this question.

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GEOMETRY (G)

CLUSTER #1: IDENTIFY AND DESCRIBE SHAPES (SQUARES, CIRCLES,
TRIANGLES, RECTANGLES, HEXAGONS, CUBES, CONES, CYLINDERS, AND
SPHERES).
This entire cluster asks students to understand that certain attributes define what a shape is
called (number of sides, number of angles, etc.) and other attributes do not (color, size,
orientation). Then, using geometric attributes, the student identifies and describes particular
shapes listed above. Throughout the year, Kindergarten students move from informal language
to describe what shapes look like (e.g., “That looks like an ice cream cone!”) to more formal
mathematical language (e.g., “That is a triangle. All of its sides are the same length”). In
Kindergarten, students need ample experiences exploring various forms of the shapes (e.g., size:
big and small; types: triangles, equilateral, isosceles, scalene; orientation: rotated slightly to the
left, „upside down‟) using geometric vocabulary to describe the different shapes. In addition,
students need numerous experiences comparing one shape to another, rather than focusing on
one shape at a time. This type of experience solidifies the understanding of the various attributes
and how those attributes are different or similar- from one shape to another. Students in
ow different-
Kindergarten typically recognize figures by appearance alone, often by comparing them to a
known example of a shape, such as the triangle on the left. For example, students in
students
Kindergarten typically recognize that the figure on the left as a triangle, but claim that the figure
on the right is not a triangle, since it does not have a flat bottom. The properties of a figure are
not recognized or known. Students make decisions on identifying and describing shapes based
decisions
on perception, not reasoning.

CCGPS.K.G.1 Describe objects in the environment using names of shapes, and describe
the relative positions of these objects using terms such as above, below, beside, in front of,
beside
behind, and next to.
This standard expects students to use positional words (such as those italicized above) to
describe objects in the environment. Kindergarten students need to focus first on location
and position of two-and-three
three-dimensional objects in their classroom prior to describing
location and position of two
two-and-three-dimension representations on paper.
dimension

CCGPS CLUSTER #2: REASON WITH SHAPES AND THEIR ATTRIBUTES.
Students compose and decompose plane or solid figures (e.g., put two triangles together to make
triangles
a quadrilateral) and build understanding of part whole relationships as well as the properties of
part-whole
the original and composite shapes. As they combine shapes, they recognize them from different
perspectives and orientations, describe their geometric attributes, and determine how they are
alike and different, to develop the background for measurement and for initial understandings of
properties such as congruence and symmetry.
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CCGPS.1.G.1 Distinguish between defining attributes (e.g., triangles are closed and three-
sided) versus non-defining attributes (e.g., color, orientation, overall size) ; build and draw
shapes to possess defining attributes.
This standard calls for students to determine which attributes of shapes are defining
compared to those that are non-defining. Defining attributes are attributes that must
always be present. Non-defining attributes are attributes that do not always have to be
present. The shapes can include triangles, squares, rectangles, and trapezoids.
Asks students to determine which attributes of shapes are defining compared to those that
are non-defining. Defining attributes are attributes that help to define a particular shape
(#angles, # sides, length of sides, etc.). Non-defining attributes are attributes that do not
define a particular shape (color, position, location, etc.). The shapes can include
triangles, squares, rectangles, and trapezoids. CCGPS.1.G.2 includes half-circles and
quarter-circles.
Example:
All triangles must be closed figures and have 3 sides. These are defining attributes.
Triangles can be different colors, sizes and be turned in different directions, so these are
non-defining.
Which figure is a triangle? How do you know this
is a tri angle?

Student 1
The figure on the left is a triangle. It has three
sides. It is also closed.

CCGPS.1.G.2 Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles,
half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular
prisms, right circular cones, and right circular cylinders) to create a composite shape, and
compose new shapes from the composite shape.
This standard calls for students to compose (build) a two-dimensional or three-
dimensional shape from two shapes. This standard includes shape puzzles in which
students use objects (e.g., pattern blocks) to fill a larger region. Students do not need to
use the formal names such as ―right rectangular prism.‖
Example:
Show the different shapes that you can make by joining a triangle with a
square.

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Show the different shapes that you can make by joining trapezoid with a
half-circle.

Show the different shapes that you can make with a cube and a
rectangular prism.

CCGPS.1.G.3 Partition circles and rectangles into two and four equal shares, describe the
shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of,
and quarter of. Describe the whole as two of, or four of the shares. Understand for these
examples that decomposing into more equal shares creates smaller shares.
This standard is the first time students begin partitioning regions into equal shares using a
context such as cookies, pies, pizza, etc... This is a foundational building block of
fractions, which will be extended in future grades. Students should have ample
experiences using the words, halves, fourths, and quarters, and the phrases half of, fourth
of, and quarter of. Students should also work with the idea of the whole, which is
composed of two halves, or four fourths or four quarters.
Example:
How can you and a friend share equally (partition) this piece of paper so that you both
have the same amount of paper to paint a picture?

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I would split the paper right down the I would split it from corner to
middle. That gives us 2 corner (diagonally). She gets half
halves. I have half of the the paper. See, if we cut here
paper and my friend has (along the line), the parts are the
the other half of the paper. same size.

Example:

Teacher: There is pizza for dinner. What Teacher: If we cut the same pizza into four
do you notice about the slices on slices (fourths), do you think the
the pizza? slices would be the same size,
larger, or smaller as the slices on
this pizza?

Student: There are two slices on the pizza.
Each slice is the same size. Those
are big slices! Student: When you cut the pizza into
fourths, the slices are smaller than
the other pizza. More slices mean
that the slices get smaller and
smaller. I want a slice of that first
pizza!

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ARC OF LESSON (OPENING, WORK SESSION, CLOSING)

“When classrooms are workshops-when learners are inquiring, investigating, and constructing-
there is already a feeling of community. In workshops learners talk to one another, ask one
another questions, collaborate, prove, and communicate their thinking to one another. The heart
of math workshop is this: investigations and inquiries are ongoing, and teachers try to find
situations and structure contexts that will enable children to mathematize their lives- that will
move the community toward the horizon. Children have the opportunity to explore, to pursue
inquiries, and to model and solve problems on their own creative ways. Searching for patterns,
raising questions, and constructing one’s own models, ideas, and strategies are the primary
activities of math workshop. The classroom becomes a community of learners engaged in
activity, discourse, and reflection.” Young Mathematicians at Work- Constructing Addition and
Subtraction by Catherine Twomey Fosnot and Maarten Dolk.

“Students must believe that the teacher does not have a predetermined method for solving the
problem. If they suspect otherwise, there is no reason for them to take risks with their own ideas
and methods.” Teaching Student-Centered Mathematics, K-3 by John Van de Walle and Lou
Ann Lovin.

Opening: Set the stage
Get students mentally ready to work on the task
Clarify expectations for products/behavior
How?
• Begin with a simpler version of the task to be presented
• Solve problem strings related to the mathematical idea/s being investigated
• Leap headlong into the task and begin by brainstorming strategies for approaching the
task
• Estimate the size of the solution and reason about the estimate
Make sure everyone understands the task before beginning. Have students restate the task in their
own words. Every task should require more of the students than just the answer.

Work session: Give ‘em a chance
Students- grapple with the mathematics through sense-making, discussion, concretizing their
mathematical ideas and the situation, record thinking in journals
Teacher- Let go. Listen. Respect student thinking. Encourage testing of ideas. Ask questions to
clarify or provoke thinking. Provide gentle hints. Observe and assess.

Closing: Best Learning Happens Here
Students- share answers, justify thinking, clarify understanding, explain thinking, question each
other
Teacher- Listen attentively to all ideas, ask for explanations, offer comments such as, “Please tell
me how you figured that out.” “I wonder what would happen if you tried…”
Anchor charts
Read Van de Walle K-3, Chapter 1
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BREAKDOWN OF A TASK (UNPACKING TASKS)

How do I go about tackling a task or a unit?

1. Read the unit in its entirety. Discuss it with your grade level colleagues. Which parts do
you feel comfortable with? Which make you wonder? Brainstorm ways to implement the
tasks. Collaboratively complete the culminating task with your grade level colleagues.
As students work through the tasks, you will be able to facilitate their learning with this
end in mind. The structure of the units/tasks is similar task to task and grade to grade.
This structure allows you to converse in a vertical manner with your colleagues, school-
wide. The structure of the units/tasks is similar task to task and grade to grade. There is a
great deal of mathematical knowledge and teaching support within each grade level
guide, unit, and task.

2. Read the first task your students will be engaged in. Discuss it with your grade level
colleagues. Which parts do you feel comfortable with? Which make you wonder?
Brainstorm ways to implement the tasks.

3. If not already established, use the first few weeks of school to establish routines and
rituals, and to assess student mathematical understanding. You might use some of the
tasks found in the unit, or in some of the following resources as beginning
tasks/centers/math tubs which serve the dual purpose of allowing you to observe and
assess.

Additional Resources:
Math Their Way: http://www.center.edu/MathTheirWay.shtml
NZMaths- http://www.nzmaths.co.nz/numeracy-development-projects-
books?parent_node=
K-5 Math Teaching Resources- http://www.k-5mathteachingresources.com/index.html
(this is a for-profit site with several free resources)
Winnepeg resources- http://www.wsd1.org/iwb/math.htm
Math Solutions- http://www.mathsolutions.com/index.cfm?page=wp9&crid=56

4. Points to remember:
• Each task begins with a list of the standards specifically addressed in that task,
however, that does not mean that these are the only standards addressed in the
task. Remember, standards build on one another, and mathematical ideas are
connected.
• Tasks are made to be modified to match your learner’s needs. If the names need
changing, change them. If the materials are not available, use what is available. If
a task doesn’t go where the students need to go, modify the task or use a different
resource.
• The units are not intended to be all encompassing. Each teacher and team will
make the units their own, and add to them to meet the needs of the learners.
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ROUTINES AND RITUALS

Teaching Math in Context and Through Problems
“By the time they begin school, most children have already developed a sophisticated, informal
understanding of basic mathematical concepts and problem solving strategies. Too often,
however, the mathematics instruction we impose upon them in the classroom fails to connect
with this informal knowledge” (Carpenter et al., 1999). The 8 Standards of Mathematical
Practices (SMP) should be at the forefront of every mathematics lessons and be the driving factor
of HOW students learn.

One way to help ensure that students are engaged in the 8 SMPs is to construct lessons built on
context or through story problems. “Fosnot and Dolk (2001) point out that in story problems
children tend to focus on getting the answer, probably in a way that the teacher wants. “Context
problems, on the other hand, are connected as closely as possible to children’s lives, rather than
to ‘school mathematics’. They are designed to anticipate and to develop children’s mathematical
modeling of the real world.”

Traditionally, mathematics instruction has been centered around a lot of problems in a single
math lesson, focusing on rote procedures and algorithms which do not promote conceptual
understanding. Teaching through word problems and in context is difficult however,
“kindergarten students should be expected to solve word problems” (Van de Walle, K-3).

A problem is defined as any task or activity for which the students have no prescribed or
memorized rules or methods, nor is there a perception by students that there is a specific correct
solution method. A problem for learning mathematics also has these features:

• The problem must begin where the students are which makes it accessible to all learners.
• The problematic or engaging aspect of the problem must be due to the mathematics
that the students are to learn.
• The problem must require justifications and explanations for answers and methods.

It is important to understand that mathematics is to be taught through problem solving. That is,
problem-based tasks or activities are the vehicle through which the standards are taught. Student
learning is an outcome of the problem-solving process and the result of teaching within context
and through the Standards for Mathematical Practice. (Van de Walle and Lovin, Teaching
Student-Centered Mathematics: K-3, page 11).

Use of Manipulatives
“It would be difficult for you to have become a teacher and not at least heard that the use of
manipulatives, or a “hands-on approach,” is the recommended way to teach mathematics. There
is no doubt that these materials can and should play a significant role in your classroom. Used
correctly they can be a positive factor in children’s learning. But they are not a cure-all that some
educators seem to believe them to be. It is important that you have a good perspective on how
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manipulatives can help or fail to help children construct ideas. We can’t just give students a ten-
frame or bars of Unifix cubes and expect them to develop the mathematical ideas that these
manipulatives can potentially represent. When a new model or new use of a familiar model is
introduced into the classroom, it is generally a good idea to explain how the model is used and
perhaps conduct a simple activity that illustrates this use. ”
(Van de Walle and Lovin, Teaching Student-Centered Mathematics: K-3, page 6).

Once you are comfortable that the models have been explained, you should not force their use on
students. Rather, students should feel free to select and use models that make sense to them. In
most instances, not using a model at all should also be an option. The choice a student makes can
provide you with valuable information about the level of sophistication of the student’s
reasoning.

Whereas the free choice of models should generally be the norm in the classroom, you can often
ask students to model to show their thinking. This will help you find out about a child’s
understanding of the idea and also his or her understanding of the models that have been used in
the classroom.

The following are simple rules of thumb for using models:
• Introduce new models by showing how they can represent the ideas for which they are
intended.
• Allow students (in most instances) to select freely from available models to use in solving
problems.
• Encourage the use of a model when you believe it would be helpful to a student having
difficulty.” (Van de Walle and Lovin, Teaching Student-Centered Mathematics: K-3,
page 8-9)
• Modeling also includes the use of mathematical symbols to represent/model the concrete
mathematical idea/thought process. This is a very important, yet often neglected step
along the way. Modeling can be concrete, representational, and abstract. Each type of
model is important to student understanding.

Use of Strategies and Effective Questioning
Teachers ask questions all the time. They serve a wide variety of purposes: to keep learners
engaged during an explanation; to assess their understanding; to deepen their thinking or focus
their attention on something. This process is often semi-automatic. Unfortunately, there are many
common pitfalls. These include:
• asking questions with no apparent purpose;
• asking too many closed questions;
• asking several questions all at once;
• poor sequencing of questions;
• asking rhetorical questions;
• asking ‘Guess what is in my head’ questions;
• focusing on just a small number of learners;
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• ignoring incorrect answers;
• not taking answers seriously.

In contrast, the research shows that effective questioning has the following characteristics:
• Questions are planned, well ramped in difficulty.
• Open questions predominate.
• A climate is created where learners feel safe.
• A ‘no hands’ approach is used, for example when all learners answer at once using mini-
whiteboards, or when the teacher chooses who answers.
• Probing follow-up questions are prepared.
• There is a sufficient ‘wait time’ between asking and answering a question.
• Learners are encouraged to collaborate before answering.
• Learners are encouraged to ask their own questions.

0-99 Chart or 1-100 Chart
(Adapted information from About Teaching Mathematics A K–8 RESOURCE MARILYN BURNS
3rd edition and Van de Walle)

Both the 0-99 Chart and the 1-100 Chart are valuable tools in the understanding of mathematics.
Most often these charts are used to reinforce counting skills. Counting involves two separate
skills: (1) ability to produce the standard list of counting words (i.e. one, two, three) and (2) the
ability to connect the number sequence in a one-to-one manner with objects (Van de Walle,
2007). The counting sequence is a rote procedure. The ability to attach meaning to counting is
“the key conceptual idea on which all other number concepts are developed” (Van de Walle, p.
122). Children have greater difficulty attaching meaning to counting than rote memorization of
the number sequence. Although both charts can be useful, the focus of the 0-99 chart should be
at the forefront of number sense development in early elementary.

A 0-99 Chart should be used in place of a 1-100 Chart when possible in early elementary
mathematics for many reasons, but the overarching argument for the 0-99 is that it helps to
develop a deeper understanding of place value. Listed below are some of the benefits of using
the 0-99 Chart in your classroom:
• A 0-99 Chart begins with zero where as a hundred’s chart begins with 1. It is important
to include zero because it is a digit and just as important as 1-9.
• A 1-100 chart puts the decade numerals (10, 20, 30, etc.) on rows without the remaining
members of the same decade. For instance, on a hundred’s chart 20 appears at the end of
the teens’ row. This causes a separation between the number 20 and the numbers 21-29.
The number 20 is the beginning of the 20’s family; therefore it should be in the
beginning of the 20’s row like in a 99’s chart to encourage students to associate the
quantities together.
• A 0-99 chart ends with the last two digit number, 99, this allows the students to
concentrate their understanding using numbers only within the ones’ and tens’ place

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values. A hundred’s chart ends in 100, introducing a new place value which may change
the focus of the places.
• The understanding that 9 units fit in each place value position is crucial to the
development of good number sense. It is also very important that students recognize that
zero is a number, not merely a placeholder. This concept is poorly modeled by a typical
1-100 chart, base ten manipulatives, and even finger counting. We have no "zero" finger,
"zero" block, or "zero" space on typical 1-100 number charts. Whereas having a zero on
the chart helps to give it status and reinforces that zero holds a quantity, a quantity of
none. Zero is the answer to a question such as, “How many elephants are in the room?”.
• Including zero presents the opportunity to establish zero correctly as an even number,
when discussing even and odd. Children see that it fits the same pattern as all of the
other even numbers on the chart.

0 1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19
20 21 22 23 24 25 26 27 28 29
30 31 32 33 34 35 36 37 38 39
40 41 42 43 44 45 46 47 48 49
50 51 52 53 54 55 56 57 58 59
60 61 62 63 64 65 66 67 68 69
70 71 72 73 74 75 76 77 78 79
80 81 82 83 84 85 86 87 88 89
90 91 92 93 94 95 96 97 98 99

While there are differences between the 0-99 Chart and the 1-100 Chart, both number charts
are valuable resources for your students and should be readily available in several places
around the classroom. Both charts can be used to recognize number patterns, such as the
increase or decrease by multiples of ten. Provide students the opportunity to explore the
charts and communicate the patterns they discover.

The number charts should be placed in locations that are easily accessible to students and
promote conversation. Having one back at your math calendar/bulletin board area provides
you the opportunity to use the chart to engage students in the following kinds of discussions.
Ask students to find the numeral that represents:
• the day of the month
• the month of the year
• the number of students in the class
• the number of students absent or any other amount relevant to the moment.

Using the number is 21, give directions and/or ask questions similar to those below.
• Name a number greater than 21.
• Name a number less than 21.
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• What number is 3 more than/less than 21?
• Is 21 even or odd?
• What numbers live right next door to 21?

Ask students to pick an even number and explain how they know the number is even. Ask
students to pick an odd number and explain how they know it is odd. Ask students to count
by 2’s, 5’s or 10’s. Tell them to describe any patterns that they see. (Accept any patterns that
students are able to justify. There are many right answers!)

Number Corner
Number Corner is a time set aside to go over mathematics skills (Standards for calendar can
be found in Social Studies) during the primary classroom day. This should be an interesting
and motivating time for students. A calendar board or corner can be set up and there should
be several elements that are put in place. The following elements should be set in place for
students to succeed during Number Corner:
1. a safe environment
2. concrete models/math tools
3. opportunities to think first and then discuss
4. student interaction
Number Corner should relate several mathematics concepts/skills to real life experiences.
This time can be as simple as reviewing the months, days of the week, temperature outside,
and the schedule for the day, but some teachers choose to add other components that
integrate more standards. Number Corner should be used as a time to engage students in a
discussion about events which can be mathematized, or as a time to engage in Number
Talks.
• Find the number ___ .
• If I have a nickel and a dime, how much money do I have? (any money combination)
• What is ___ more than ___?
• What is ___ less than ___?
• Mystery number: Give clues and they have to guess what number you have.
• This number has ___tens and ___ ones. What number am I?
• What is the difference between ___ and ____?
• What number comes after ___? before ___?
• Tell me everything you know about the number ____. (Anchor Chart)

Number Corner is also a chance to familiarize your students with Data Analysis. This
creates an open conversation to compare quantities, which is a vital process that must be
explored before students are introduced to addition and subtraction.

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• At first, choose questions that have only two mutually exclusive answers, such as yes
or no (e.g., Are you a girl or a boy?), rather than questions that can be answered yes,
no, or maybe (or sometimes). This sets up the part-whole relationship between the
number of responses in each category and the total number of students present and it
provides the easiest comparison situation (between two numbers; e.g., Which is
more? How much more is it?). Keep in mind that the concept of less than (or fewer)
is more difficult than the concept of greater than (or more). Be sure to frequently
include the concept of less in your questions and discussions about comparisons.
• Later, you can expand the questions so they have more than two responses. Expected
responses may include maybe, I’m not sure, I don’t know or a short, predictable list
of categorical responses (e.g., In which season were you born?).
• Once the question is determined, decide how to collect and represent the data. Use a
variety of approaches, including asking students to add their response to a list of
names or tally marks, using Unifix cubes of two colors to accumulate response
sticks, or posting 3 x 5 cards on the board in columns to form a bar chart.
• The question should be posted for students to answer. For example, “Do you have an
older sister?” Ask students to contribute their responses in a way that creates a
simple visual representation of the data, such as a physical model, table of responses,
bar graph, etc.
• Each day, ask students to describe, compare, and interpret the data by asking
questions such as these: “What do you notice about the data? Which group has the
most? Which group has the least? How many more answered [this] compared to
[that]? Why do you suppose more answered [this]?” Sometimes ask data gathering
questions: “Do you think we would get similar data on a different day? Would we
get similar data if we asked the same question in another class? Do you think these
answers are typical for first graders? Why or why not?”
• Ask students to share their thinking strategies that justify their answers to the
questions. Encourage and reward attention to specific details. Focus on relational
thinking and problem solving strategies for making comparisons. Also pay attention
to identifying part-whole relationships; and reasoning that leads to interpretations.
• Ask students questions about the ideas communicated by the representation used.
What does this graph represent? How does this representation communicate this
information clearly? Would a different representation communicate this idea better?
• The representation, analysis, and discussion of the data are the most important parts
of the routine (as opposed to the data gathering process or the particular question
being asked). These mathematical processes are supported by the computational
aspects of using operations on the category totals to solve part-whole or “compare”
problems.

Number Talks
Number talks are a great way for students to use mental math to solve and explain a variety of
math problems. A Number Talk is a powerful tool for helping students develop computational
fluency because the expectation is that they will use number relationships and the structures of
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numbers to add, subtract, multiply and divide. A Number Talk is a short, ongoing daily routine
that provides students with meaningful ongoing practice with computation. Number Talks should
be structured as short sessions alongside (but not necessarily directly related to) the ongoing
math curriculum. A great place to introduce a Number Talk is during Number Corner. It is
important to keep Number Talks short, as they are not intended to replace current curriculum
or take up the majority of the time spent on mathematics. In fact, teachers need to spend only
5 to 15 minutes on Number Talks. Number Talks are most effective when done every day. As
prior stated, the primary goal of Number Talks is computational fluency. Children develop
computational fluency while thinking and reasoning like mathematicians. When they share their
strategies with others, they learn to clarify and express their thinking, thereby developing
mathematical language. This in turn serves them well when they are asked to express their
mathematical processes in writing. In order for children to become computationally fluent, they
need to know particular mathematical concepts that go beyond what is required to memorize
basic facts or procedures.

Students will begin to understand major characteristics of number, such as:
• Numbers are composed of smaller numbers.
• Numbers can be taken apart and combined with other numbers to make new numbers.
• What we know about one number can help us figure out other numbers.
• What we know about parts of smaller numbers can help us with parts of larger numbers.
• Numbers are organized into groups of tens and ones (and hundreds, tens and ones and so forth).
• What we know about numbers to 10 helps us with numbers to 100 and beyond.
All Number Talks follow a basic six-step format. The format is always the same, but the
problems and models used will differ for each number talk.
1. Teacher presents the problem. Problems are presented in many different ways: as dot
cards, ten frames, sticks of cubes, models shown on the overhead, a word problem or a
written algorithm.
2. Students figure out the answer. Students are given time to figure out the answer. To
make sure students have the time they need, the teacher asks them to give a “thumbs-up”
when they have determined their answer. The thumbs up signal is unobtrusive- a message
to the teacher, not the other students.
3. Students share their answers. Four or five students volunteer to share their answers and
the teacher records them on the board.
4. Students share their thinking. Three or four students volunteer to share how they got
their answers. (Occasionally, students are asked to share with the person(s) sitting next to
them.) The teacher records the student's thinking.
5. The class agrees on the "real" answer for the problem. The answer that together the
class determines is the right answer is presented as one would the results of an
experiment. The answer a student comes up with initially is considered a conjecture.
Models and/or the logic of the explanation may help a student see where their thinking
went wrong, may help them identify a step they left out, or clarify a point of confusion.
There should be a sense of confirmation or clarity rather than a feeling that each problem
is a test to see who is right and who is wrong. A student who is still unconvinced of an
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answer should be encouraged to keep thinking and to keep trying to understand. For some
students, it may take one more experience for them to understand what is happening with
the numbers and for others it may be out of reach for some time. The mantra should be,
"If you are not sure or it doesn't make sense yet, keep thinking."
6. The steps are repeated for additional problems.
Similar to other procedures in your classroom, there are several elements that must be in place to
ensure students get the most from their Number Talk experiences. These elements are:
1. A safe environment
2. Problems of various levels of difficulty that can be solved in a variety of ways
3. Concrete models
4. Opportunities to think first and then check
5. Interaction
6. Self-correction

Mathematize the World through Daily Routines
The importance of continuing the established classroom routines cannot be overstated. Daily
routines must include such obvious activities such as taking attendance, doing a lunch count,
determining how many items are needed for snack, lining up in a variety of ways (by height, age,
type of shoe, hair color, eye color, etc.), daily questions, 99 chart questions, and calendar
activities. They should also include less obvious routines, such as how to select materials, how to
use materials in a productive manner, how to put materials away, how to open and close a door,
how to do just about everything! An additional routine is to allow plenty of time for children to
explore new materials before attempting any directed activity with these new materials. The
regular use of the routines are important to the development of students’ number sense,
flexibility, and fluency, which will support students’ performances on the tasks in this unit.

Workstations and Learning Centers
It is recommended that workstations be implemented to create a safe and supportive environment
for problem solving in a standards based classroom. These workstations typically occur during
the “exploring” part of the lesson, which follows the mini-lesson. Your role is to introduce the
concept and allow students to identify the problem. Once students understand what to do and you
see that groups are working towards a solution, offer assistance to the next group.

Groups should consist of 2-5 students and each student should have the opportunity to work with
all of their classmates throughout the year. Avoid grouping students by ability. Students in the
lower group will not experience the thinking and language of the top group, and top students will
not hear the often unconventional but interesting approaches to tasks in the lower group (28, Van
de Walle and Lovin 2006).

In order for students to work efficiently and to maximize participation, several guidelines must
be in place (Burns 2007):
1. You are responsible for your own work and behavior.
2. You must be willing to help any group member who asks.
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3. You may ask the teacher for help only when everyone in your group has the same question.

These rules should be explained and discussed with the class so that each student is aware of the
expectations you have for them as a group member. Once these guidelines are established, you
should be able to successfully lead small groups, which will allow you the opportunity to engage
with students on a more personal level while providing students the chance to gain confidence as
they share their ideas with others.

The types of activities students engage in within the small groups will not always be the same.
Facilitate a variety of tasks that will lead students to develop proficiency with numerous concepts
and skills. Possible activities include: math games, related previous Framework tasks, problems,
and computer-based activities. With all tasks, regardless if they are problems, games, etc. include
a recording sheet for accountability. This recording sheet will serve as a means of providing you
information of how a child arrived at a solution or the level at which they can explain their
thinking (Van de Walle 2006).

Games
“A game or other repeatable activity may not look like a problem, but it can nonetheless be
problem based. The determining factor is this: Does the activity cause students to be reflective
about new or developing relationships? If the activity merely has students repeating procedure
without wrestling with an emerging idea, then it is not a problem-based experience. However the
few examples just mentioned and many others do have children thinking through ideas that are
not easily developed in one or two lessons. In this sense, they fit the definition of a problem-
based task.

Just as with any task, some form of recording or writing should be included with stations
whenever possible. Students solving a problem on a computer can write up what they did and
explain what they learned. Students playing a game can keep records and then tell about how
they played the game- what thinking or strategies they used.” (Van de Walle and Lovin,
Teaching Student-Centered Mathematics: K-3, page 26)

Journaling

"Students should be writing and talking about math topics every day. Putting thoughts into words
helps to clarify and solidify thinking. By sharing their mathematical understandings in written
and oral form with their classmates, teachers, and parents, students develop confidence in
themselves as mathematical learners; this practice also enables teachers to better monitor student
progress." NJ DOE

"Language, whether used to express ideas or to receive them, is a very powerful tool and should
be used to foster the learning of mathematics. Communicating about mathematical ideas is a way
for students to articulate, clarify, organize, and consolidate their thinking. Students, like adults,
exchange thoughts and ideas in many ways—orally; with gestures; and with pictures, objects,
and symbols. By listening carefully to others, students can become aware of alternative
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